Weber's class number problem and p--rationality in the cyclotomic Z^-extension of Q
Résumé
Let K:=Q(ℓ^n), n≥0, be the nth layer in the cyclotomic Z_ℓ-extension of Q. It is conjectured that, for all ℓ and n, K is principal (especially for ℓ=2, a conjecture due to Weber). Many studies (Ichimura--Morisawa--Nakajima--Okazaki...) go in this direction, as the Miller use of the Cohen--Lenstra--Martinet heuristics. Nevertheless, we examine in what circumstances a counterexample may be possible. For this, computations show that the p-torsion group T_K of the Galois group of the maximal abelian p-ramified pro-p-extension of K is not always trivial. This questions the relevance of the conjecture since #T_K=#C_K # R_K #W_K, where C_K is the p-class group of K, R_K its normalized p-adic regulator, #W_K=1 for p>2, #W_K=2^(# {v,v | 2}-1) for p=2; nevertheless, no counterexample has been found so far, even using the reflection theorem. When n increases, some relative components T_K^* may appear for large p. We give a method (Theorem 4.6), for testing #T_K≠1, allowing larger values of ℓ^n than those of the literature. Finally, we consider the subfields K of the composite Q^ of the Z_ℓ-extension and give programs finding again some rare cases of non-trivial class groups (Fukuda--Komatsu--Horie) due to genus theory in connection with a deep link involving R_K (Theorem 6.2) in relation with Greenberg's conjecture as initiated, via p-adic zeta-functions, by Taya. In all attempts, Jaulent's logarithmic class group Cl_K, K
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https://hal.science/hal-02935539
Soumis le : jeudi 10 septembre 2020-14:17:45
Dernière modification le : lundi 11 mars 2024-14:44:05
Archivage à long terme le : jeudi 3 décembre 2020-02:05:01
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Georges Gras. Weber's class number problem and p--rationality in the cyclotomic Z^-extension of Q. 2020. ⟨hal-02935539v1⟩
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